9.4 #18

[Riccardo R.]

In this problem I'm not sure why I should have a particular solution
with both sine and cosine terms. My Fourier series for F(t), since
F(t) is odd, has obviously only sine terms. Shouldn't x_sp be the same
as F(t), but with a different constant in front of sine and cosine
terms?

Thanks.

Riccardo.


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[Professor Laugesen]

The differential equation has x'(t) in it. That term flips cosines to
sines, and sines to cosines. That's why you need both sine and cosine
in your series guess for x_sp(t).

Another way of saying it is that when you write down a Fourier series
for x_sp(t) with unknown coefficients, you are really using the Method
of Undetermined Coefficients - and in that method, if the right side
of the DE has a sine or a cosine then your guess for the particular
solution should involve *both* a sine and a cosine.